cp's OEIS Frontend

A225432 Twice the coefficient of sqrt(q) in e^h, where e is the fundamental unit and h is the class number of Q(sqrt(q)), q prime and congruent to 1 mod 4. (The coefficient lies in (1/2)Z, so twice it is an integer.)

%I A225432 #41 Jan 15 2025 19:44:53
%S A225432 1,1,2,1,2,10,1,5,250,106,1138,2,25,146,298,5,17,1,97,253970,2,226,
%T A225432 3034,9148450,2050,10,157,126890,1,14341370,5,110671282,986,7586,530,
%U A225432 130,173,5129602,11068353370,21685,694966754,17883410,5528222698,17,41,11248618,60037,10,242718010,24514292738
%N A225432 Twice the coefficient of sqrt(q) in e^h, where e is the fundamental unit and h is the class number of Q(sqrt(q)), q prime and congruent to 1 mod 4. (The coefficient lies in (1/2)Z, so twice it is an integer.)
%C A225432 This also arises in the relation satisfied by Euler classes in the connective K-theory of the classifying space of the group of order pq, where p=(q-1)/2. See p. 39 in Bruner and Greenlees, cited below. Take an irreducible representation of the cyclic group of order q which generates the representations as a ring, induce it up to the group of order pq, and let z be its Euler class in ku^{2p}(BG_{pq}). Then z satisfies the relation z^3 -2bq z^2 + qz = 0. This follows from the arithmetic fact that in Q(sort(q)) we have the relation e^h = a + b sqrt(q), as shown on pp. 39-42 of Bruner and Greenlees.
%C A225432 This is closely related to the subsequence of A078357 containing those entries such that the corresponding entry in A077426 is prime. However, a(22) = 226 (corresponding to e^3 = 1710 + 113*sqrt(229)) does not occur in A078357, and more such terms appear after this.
%C A225432 For the n-th Pythagorean prime q=A002144(n), a(n) is also -1/q of the coefficient of term x in the minimal polynomial of A=Product_{a} 2*sin(a*Pi/q) (where the index runs through all quadratic residues in {1,2,...,q-1}) and B=Product_{b} 2*sin(b*Pi/q) (where the index runs through all quadratic nonresidues in {1,2,...,q-1}). It is easy to show that A*B = p. By the class number formula of real quadratic number fields, one obtains B/A = e^(+-2h), so A+B = sqrt(q)*(e^h+e^(-h)) is exactly q*a(n). - _Zichang Wang_, Dec 15 2022
%D A225432 R. R. Bruner and J. P. C. Greenlees, The Connective K-theory of Finite Groups, Memoirs AMS, Vol. 165, No. 785, 2003.
%D A225432 T. Mitsuhiro, T. Nakahara and T. Uehara, The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit, Canadian Mathematical Bulletin, 38(1)(1995), 98-103.
%H A225432 Zichang Wang, <a href="/A225432/b225432.txt">Table of n, a(n) for n = 1..2000</a>
%H A225432 R. R. Bruner and J. P. C. Greenlees, <a href="https://www.semanticscholar.org/paper/The-Connective-K-Theory-of-Finite-Groups-Bruner-Greenlees/ad7f06aae2f629052f1472f94959ba941a4835c6">The Connective K-theory of Finite Groups</a>, Semantic Scholar.
%H A225432 T. Mitsuhiro, T. Nakahara and T. Uehara, <a href="https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/class-number-formula-of-a-real-quadratic-field-and-an-estimate-of-the-value-of-a-unit/693D31C6D873028486E2D1A32ED34970">The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit</a>.
%t A225432 (* e.g., first 270 terms *)
%t A225432 Lq = Select[4*Range[1000] + 1, PrimeQ[#] &];
%t A225432 Lh = NumberFieldClassNumber[Sqrt[Lq]];
%t A225432 Le = NumberFieldFundamentalUnits[Sqrt[Lq]];
%t A225432 Transpose[RootReduce[(Le^(2 Lh) + 1)/(Sqrt[Lq] Le^Lh)]][[1]]
%t A225432 (* _Zichang Wang_, Dec 15 2022 *)
%o A225432 (Magma)
%o A225432 // Magma code to generate all terms for which the prime q is less than or equal to 4N+1 (an initial segment of the sequence). (Note that the brute force computation of the fundamental unit is very inefficient, and will have trouble producing the 39th term.)
%o A225432 N := 40;
%o A225432 pr := [4*n+1 : n in [1..N] | IsPrime(4*n+1)];
%o A225432 for i in [1..#pr] do
%o A225432    q := pr[i];
%o A225432    Q<s> := QuadraticField(q);
%o A225432    h := ClassNumber(Q);
%o A225432    x := 1;
%o A225432    while not IsSquare(x*x*q-4) do
%o A225432       x := x+1;
%o A225432    end while;
%o A225432    x := x/2;
%o A225432    tr,y := IsSquare(x*x*q-1);
%o A225432    e := y + x*s;
%o A225432    eh := e^h;
%o A225432    b := (eh-Trace(eh)/2)/s;
%o A225432    print i,2*b;
%o A225432 end for;
%Y A225432 Cf. A002144, A077426, A078357.
%K A225432 nonn
%O A225432 1,3
%A A225432 _Robert R. Bruner_, May 07 2013
%E A225432 a(39) onward from _Zichang Wang_, Dec 15 2022